Oscillations of longitudina resistivity =Shubnikov-deHaas, minima close to 0 .

1. K. v. Klitzing discovery: 1980 Nobel prize: 1985 Oscillations of longitudina resistivity =Shubnikov-deHaas, minima close to 0. Plateaux in Hall resistivity r=h/(ne2) with integer n correspond to the minima (From Datta page 25)

2. Origin of the Oscillations of longitudina resistivity =Shubnikov-deHaas r(w) with H r(w) without H E E resistivity minima close to 0

3. Rectangular conductor very thin in z direction uniform in x direction confined in y direction with B in z direction. Assuming for the sake of argument that H is separable, the transverse dimensions yield infinite solutions that are called subbands: let us assume that only one z subband is occupied and the confinement along y is described by U(y).

4. For n=integer,..filled or empty LL  gap, no scattering  xx resistivity =0 For n=0.5,1.5,2.5,.. Half filled LL  maximum xx resistivity r(w) with H r(w) with H Thisis a paradox: onewouldexpect minimum resistancewhen LL isat Fermi energy, butitis maximum; howdoes the sample carry the currentwhen EFisbetween LL and there are no statesat EF? Reply: there are 1d statesat the eges of the sample (hedge states) thatcarry the current! The minimum resistivityisverylowbecause of the suppression of relaxation. Carriersthat go to opposite directions are far away and nevermeet. E E but more like this Due to impurities, the DOS isnotlikethis Due to disorder and impurities, it is possible to find the Fermi level away from the LL (otherwise it is unlikely to find it where the DOS is small and a small charge can move EF). The conduction takes place through the M hedge states that have very small resistance.

5. Origin of zero resistance(see also Datta page 175) y kx We try to include confining potential U(y) along y as a perturbation, which is nearly constant over the extent of the LL wavefunction. Including confining potential in first order, states with k in x direction in the LL n have energy: In the middle of the sample, bulk-like eigenvalues and eigenvectors prevail, but near the edges the levels are shifted by U yielding a quasi-continuum of levels, also at fermi energy. Current in edge state can be evaluated by the group velocity:

6. No backscattering takes place.This situation when EF is between two LL, otherwise the LL at Fermi level carries current within the sample with scattering and maximum resistance (not an explanation but a description) 6

7. Could be measured i n balisticconductorswithin a few % in QHE with betterthanppbaccuracy! in QHE mm-sized electron mean free paths because curent carrying states in opposite directions are localized on opposite sides  no backscattering

8. Quantizedresistance

9. Convincingexplanation ? NO! Drude theoryiarough, the resultisextremely precise, and why the plateaux? Elaborated from a seminar by Michael Adler

10. Due to the applied bias, a current flows in tha sample; this produces the Hall field in the y direction. So the upper edge states, where electrons go to the left , have the chemicel potential mR of the right electrode, the lower ones have the chemical potential mL of the left electrode. The potential drop VLongitudina along x for both is zero. Since the potential drop along y is VH, mR-mL=VH. Good, but a seriousdoubtremains. Wedidseveral crude approximations. Whyis the result so precise)? 10

11. Why the plateaux? Why so exact? y x Consider a Metal ribbon long side along x, magneticfieldalong z. 11

12. Laughlindoesnoteveninsertconfiningpotential U(y) whichplays no role in hisargument. Nowadd an electricfieldalong y.

14. Considerclosingitas a ring pierced by a flux, with opposite sidesconnectedto chargereservoirs of infinite capacity , eachas the samepotentialas the side to whichitisconnected. A current I flowsaround, the Hall potential VHexistsbetweenreservoirs.

15. Next replace E by a time-dependent flux f inside; it produces a e.m.f that excites a current I around (along x) but the magneticfieldalong z thenproduces a Hall electricfield VH, along the ribbon, thatwill transfer charge q from onereservoir to the other, contributingqVH to the energy. If the flux is one fluxon, the system is completely restored in previous state .

16. In order to have the ribbon in same state as before the fluxon is applied, q=ne with n integer. Hence, Laughlin argues that the same holds true even in the presence of interactions. Laughlin’sargumenthasbeencriticized on the groundsthatdifferentcycles of the pumpmaytransportdifferentamounts of charge, since q isnot a conservedquantity. Itis the meantransferredchargethat must be quantized. Itappears to me that the criticismisrathersophistic, becauseif the averageis an exactintegerthatdoesimplythateverymeasurementgives an integer. Onecouldenvisage a situation wheretwoexactlyequallylikelyoutcomes are 0.80000 and 1.20000 (sharp!) and so the averageis 1,0000 withouthavingintegeroutcomeseach time. Oneshoulddiscoverfractionallychargedrealelectronsbeforeacceptingthisexplanation. However some authors of the abovecriticismshaveproduced a remarkable alternative explanation.

17. A According to the Authors, Laughlin’sargumentis short in oneimportantstep, namely, the inclusion of topological quantum numbers.

19. 19

20. Chern formula Gauss and Charles Bonnet formula 20

21. The QHE is the prototype topological insulator Topological Insulators • (band) insulator with a nonzero gap to excitated states • topological number stable against any (weak) perturbation • gapless edge mode • Low-energyeffectivetheory of topologicalinsulators = topologicalfieldtheory (Chern-Simons) Bi2Se3 is a 3d topological insulator http://www.riken.go.jp/lab-www/theory/colloquium/furusaki.pdf http://wwwphy.princeton.edu/~yazdaniweb/Research_TopoInsul.php

22. Fractional Quantum Hall effect D.C. Tsui, H.L. Störmer and A.C. Gossard, prl (1982): quantization of Hall conductanceatn= 1/3 and 2/3 below 1 Kelvin The fractional quantum Hall effect (FQHE) is a physical phenomenon in which a certain strongly correlated system at T under a very strong magnetic field behaves as if it were composed of particles with fractional charge (1998 Nobel Prize).

23. experiments performed on gallium arsenideheterostructures

24. Denominators almost ever odd. The FQHE is a different phenomenon, requiring a different explanation. It is believed that the effect is due to the Coulomb interaction. All electrons in LLL treating interactions as a perturbation that tends to lower the symmetry.

25. Wave functions for the LLL in polar coordinates 25

26. LaughlinsoughttheeMany-Body wavefunction for LLL in the form We want it to be an eigenfunction of total angular momentum

27. Physical Picture: Outer electron encloses lmax fluxons R. Laughlin

28. Letusevaluate q. Expandingthe product in Laughlinansatz the maximum power of z of anyparticleevidentlyturns out to be: lmax= Nelq In terms of such wavefunctions one can estimate correlations and compute successfully the relevant quantities. Quasiparticles of fractional charge e/q obey anyon statistics (exchange of two quasiparticles brings a phase pn, which can be calculated as a Berry phase). Very accurate wave function when compared with numerical estimates. q=1 yields integer QHE wavefunction 28

29. From a seminar by Michael Adler

30. Some CONCLUSIONS The dimensionalityconditions the many-body behaviouralreadyat the classicallevel- e.g. phasetransitions At the quantum level, the transportproperties are verydifferentwhennanoscopicobjects are considered, and thisrequires new intriguingconcepts and funnymathematicalmethods, many of which involve the Berry phasetoo. Here I just recall some. The subjectis in rapidevolution, and new applications are also under way. Ballisticconduction, nonlinearmagneticbehaviour, possibility of variouskinds of pumping. The role of correlationeffectsismuch more critical in 2d (e.g. QHE, chargefractionalization, anyons , Kosterlitz-Thouless ) and aboveall in 1d (Peierlstransition and chargefractionalization, ) and therewas no time to introduce others, thatwouldrequire the Luttinger Liquid formalism….

31. holon spinon Charge-spin separation spins are assumed to jump freely to empty sites 31

32. spinon propagation holon Bosonization holon spinon nearby sites are assumed to exchange spins holon also travels Fermi particle becomes a pair of boson excitations that can propagate with different speeds (in 1d only) Fermion operators can be expressed in terms of bosons: spinons and holons and problem is solved 32

33. Separation Realized experimentally! 33